Determinant of a 4x4 Matrix

The matrix determinant is a special number associated with any square matrix. The meaning of the determinant is the scale factor for the measure when the matrix is regarded as a linear transformation. The determinant of a 4 x 4 matrix represents the use of 4 x 4 matrix for the basic operations and in the calculation of the determinant value.

(Source from Wikipedia)

 

Examples to explain "determinant of a 4x4 matrix "

 

  • Calculate the determinant value for the 4 x 4 matrix  `[[1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1]]`   

Solution:

It is in the form of `[[g_11,g_12,g_13,g_14],[g_21,g_22,g_23,g_24],[g_31,g_32,g_33,g_34],[g_41,g_42,g_43,g_44]]`   

Let us take it as the determinant G = `|[1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1]|` 

We have to expand it in the first row itself

G = `g_11 `  `[[g_22,g_23,g_24],[g_32,g_33,g_34],[g_42,g_43,g_44]]``g_12 `  `[[g_21,g_23,g_24],[g_31,g_33,g_34],[g_41,g_43,g_44]]` +  `g_13 `  `[[g_21,g_22,g_24],[g_31,g_32,g_34],[g_41,g_42,g_44]]` - `g_14 `  `[[g_21,g_22,g_23],[g_31,g_32,g_33],[g_41,g_42,g_43]]`                                                      

G = 1`|[1,1,1],[1,1,1],[1,1,1]|`   -  1`|[1,1,1],[1,1,1],[1,1,1]|`    + 1 `|[1,1,1],[1,1,1],[1,1,1]|`     -1`|[1,1,1],[1,1,1],[1,1,1]|` 

The value of the determinant    `|[1,1,1],[1,1,1],[1,1,1]|`

= 1(1-1) - 1(1-1) +1(1-1)

=0

G = 1(0) - 1(0) + 1(0) -1(0)

G = 0 determinant value

We can also say directly its determinant value will be zero, because by the property of the determinant the identica rows or columns in the determinant gets the determinnant value to be zero. Here the four rows are identical.

 

 

problems to explain "determinant of a 4x4 matrix "

 

  • Calculate the determinant value for the 4 x 4 matrix  `[[1,2,3,4],[5,1,6,1],[7,1,8,1],[3,1,4,1]]`   

Solution:

It is in the form of `[[g_11,g_12,g_13,g_14],[g_21,g_22,g_23,g_24],[g_31,g_32,g_33,g_34],[g_41,g_42,g_43,g_44]]`   

Let us take it as the determinant G =  `|[1,2,3,4],[5,1,6,1],[7,1,8,1],[3,1,4,1]|` 

We have to expand it in the first row itself

G = `g_11 `  `[[g_22,g_23,g_24],[g_32,g_33,g_34],[g_42,g_43,g_44]]``g_12 `  `[[g_21,g_23,g_24],[g_31,g_33,g_34],[g_41,g_43,g_44]]` +  `g_13 `  `[[g_21,g_22,g_24],[g_31,g_32,g_34],[g_41,g_42,g_44]]` - `g_14 `  `[[g_21,g_22,g_23],[g_31,g_32,g_33],[g_41,g_42,g_43]]`                                                      

G = 1`|[1,6,1],[1,8,1],[1,4,1]|`   -  2`|[5,6,1],[7,8,1],[3,4,1]|`     + 3 `|[5,1,1],[7,1,1],[3,1,1]|`     -4`|[5,1,6],[7,1,8],[3,1,4]|`  

The value of the determinant   `|[1,6,1],[1,8,1],[1,4,1]|` 

= 1(8-4) - 6(1-1) +1(4-8)

=0

The value of the determinant  `|[5,6,1],[7,8,1],[3,4,1]|`

= 5(8-4) - 6(7-3) +1(28-24)

=20 - 24 + 4

=0

The value of the determinant `|[5,1,1],[7,1,1],[3,1,1]|`

= 5(1-1) - 1(7-3) +1(7-3)

= -4 +4 =0

The value of the determinant `|[5,1,6],[7,1,8],[3,1,4]|`

= 5(4-8) - 1(28-24) +6(7-3)

= -20 - 4 + 24

=0

G = 1(0) - 2(0) + 3(0) -4(0)

G = 0 determinant value